new visual secret sharing schemes using probabilistic method

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New visual secret sharing schemes using probabilistic method

Ching-Nung Yang

Pattern Recognition Letters 25 , 2004

指導老師：李南逸Speaker ：黃資真

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Outline

Introduction ProbVSS scheme

A k-out-of-n ProbVSS scheme A 2-out-of-2 ProbVSS scheme A 2-out-of-n ProbVSS scheme A k-out-of-k ProbVSS scheme A general k-out-of-n ProbVSS scheme

Conclusion

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Introduction

White pixels show the contrast of recovered image.

The new scheme is non-expansible shadow.

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ProbVSS schemen shadows

k Get the shared secret by stacking their shadows.

n*1 Boolean matrices of sharing a white pixel.

n*1 Boolean matrices of sharing a black pixel.

L(V)

‘OR-ed’ operation of this k-tuple column vector V.

λ L( ) values

γ L( ) values

α contrast α>0

The threshold probability 0 1≦ ≦The appearance probability of white pixel in the write area.

The appearance probability of white pixel in the black area.

0C

1C

Probabilistic scheme use the abbreviation ProbVSS scheme.

THp

0p

1p

THp

0C

1C

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A k-out-of-n ProbVSS scheme

A (k,n) ProbVSS Scheme is conditions ：

n*1 matrices in the set and , L(V) operation values of all matrices form two sets λ and γ.

The two sets λ and γ satisfy that and≧ ≦ -α.

For any subset { } of {1,2,…,n} with q<k, the and are the same.

0C 1C

THp0p

1p THp

1 2, ,..., qi i i 0p

1p

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A 2-out-of-2 ProbVSS scheme

Construction ：

Theorem ：

Proof.

, denotes the set of all n*1 column matrices.

ex

1 0 1

1 , 1 , 0

0 1 1

i j

2,0

3*1：

0 0,0 2,0

1 1,1

0 1

{ , }

={ }

C and C is sets consisting

of 2*1 matrices.

C

C

TH p = 0.5

= 0.5

0 10 0,0 2,0 0 1

0 11 1,1 1 0

0 10 1 0

0 11 0 1

{ , }={ }

={ }={ }

so

{ ( ) ( )}={0,1} , p 0.5

{ ( ) ( )}={1,1} , p 0

C

C

L L

L L

，

，

，

，

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A 2-out-of-2 ProbVSS scheme

Proof of third condition “security” ：

Shadow 1 λ={L([0]) , L([1])} = {0,1} , =0.5

γ={L([0]) , L([1])} = {0,1} , =0.5

Shadow 2 λ={L([0]) , L([1])} = {0,1}, =0.5

γ={L([1]) , L([0])} = {1,0}, =0.5

0p

0p

1p

1p

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A 2-out-of-n ProbVSS scheme Construction 1 ：

Theorem 1 ：

0 0,0 ,0

1 [ / 2],1

1 [ / 2],1 [ / 2] 1,1

0 1

{ }

{ } (even n)

C { } (odd n)

C and C is sets consisting of n*1 matrices

n

n

n n

C

C

，

　　　

，

THp 0.5

(even n)4 4n+1

(odd n)4

n

n

n

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A 2-out-of-n ProbVSS scheme Proof ：　 A (2,3) ProbVSS scheme

0 0,0 3,0

1 1,1 2,1

0

{ }= 0 1

0 1

C { }

0 0 1 0

= 0 1 0 1 1 0

1 0 0 0 1 1

0 1,

0 1

C

L L

1

，，

，

1 1

　，，，，，

0 1

TH

0,1

0 0 1, , ,

0 1 0

1 1 0, ,

1 0 1

{0,1,1,1,1,1}

p =1/2 p =1/6

p =0.5 =1/3

L L L

L L L

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0 0,0 4,0

2 2,1

0 1

0 1{ }=

0 1

0 1

C { }

1 0 1 0 0 1

1 0 0 1 1 0 =

0 1 1 0 1 0

0 1 0 1 0 1

0

0

C

L

，，

　，，，，，

0 1

TH

1, 0,1

1

1 0 1, , ,

1 0 0

0 1 0, ,

1 0 1

{1,0,1,1,1,1}

p =1/2 p =1/6

p =0.5 =1/3

L

L L L

L L L

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A 2-out-of-n ProbVSS scheme Construction 2 ：

Theorem 2 ：

n-1

0 0,0 ,0 ,0

1 -1,1

0 1

{ ... }

{ }


n n

n

C

C

，

THp 1/

1/

n

n

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0 0,0 3,0 3,0

1 2,1

0

{ }= 0 1 1

0 1 1

C { }

0

= 1 1 0

0 1 1

0 1 1, , 0,

0 1 1

C

L L L

1 1

，，，，

1 1

　，，

0 1

TH

1,1

1 1 0, ,

1 0 1

{1,1,1}

p =1/3 p =0

p =1/3 =1/3

L L L

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A k-out-of-k ProbVSS scheme

0 ,0

1 ,1

0 1

C , even and 0 i k

C , odd and 0 i k


i

i

where i is

where i is

Construction 2 ：

Theorem 2 ：k-1

TH

k-1

p =1/2

=1/2

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A k-out-of-k ProbVSS scheme Proof ：　 A (3,3) ProbVSS scheme

0 0,0 2,0

1 1,1 3,1

0 0

{ }= 0 1 0 1

0 0 1 1

1 0 0 1

C { } = 0 1 0 1

0 0 1 1

0

0 1

0 0

C

L L

1 1

，，，，

，，，，

1

， 0

0 1 0,1,1,1

1 1

1 0 0 1

0 1 0 1 {1,1,1,1

0 0 1 1

L L

L L L L

1

，，

，，，

0 1

TH

}

p =1/4 p =0

p =1/4 =1/4

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A general k-out-of-n ProbVSS schemeh The ‘whiteness’ of white pixel

l The ‘whiteness’ of black pixel

m shadow size

n*m Boolean matrices of sharing a white pixel.

n*m Boolean matrices of sharing a black pixel.

T(.) T(.) is transferred to a set of ‘m’ n*1 column matrices.

0B

1B

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A general k-out-of-n ProbVSS scheme Construction :

Theorem ：

0 0

1 1

C ( )

C ( )

T B

T B

THph

mh l

m

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A general k-out-of-n ProbVSS scheme0 1

0 0 1 1 1 0 1 1 0 0 0 1

0 0 1 1 0 1 1 1 0 0 1 0(1)

0 0 1 0 1 1 1 1 0 1 0 0

0 0 0 1 1 1 1 1 1 0 0 0

B B

0 0

1 1

0 0 1 1 1 0

0 0 1 1 0 1(2) C ( ) , , , , ,

0 0 1 0 1 1

0 0 0 1 1 1

1 1 0 0

1 1 0 0 C ( ) , , ,

1 1 0

1 1 1

T B

T B

0 1

1 0, ,

1 0 0

0 0 0

0 0 1

0 , 0 , 1 ,

0 0 1(3) 1,1,0,1,1,1

1 1 0

1 , 0 , 1

0 1 1

L L L

L L L

1 1 0

1 , 1 , 0 ,

1 1 01,1,0,1,1,1

0 0 1

0 , 1 , 0

1 0 0

L L L

L L L

Proof ： A Shamir’s (3,4) VSS scheme with white and black matrices

0 1(4) 1/ 3 1/ 6

1/ 3 =1/6TH

p p

p

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Conclusion

New (k,n) ProbVSS schemes with non-expansible shadow size based on the probabilistic method.

The conventional VSS scheme can be transferred to ProbVSS scheme.

The ProbVSS scheme is a different view of the conventional VSS scheme.

new visual secret sharing schemes using probabilistic method

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